When the circumference of a toy balloon is increased from $20\text{ cm}$ to $25\text{ cm}$, its radius is increased by

### Concept & Circumference of a Circle The circumference of a circle is directly proportional to its radius. The formula relating them is: $$C = 2\pi r$$ Therefore, any change in the circumference ($\Delta C$) can be directly translated to a change in the radius ($\Delta r$) using $\Delta r = \frac{\Delta C}{2\pi}$. ### Step-by-Step Solution * **Given:** Initial circumference $C_1 = 20\text{ cm}$, Final circumference $C_2 = 25\text{ cm}$. * **Deduction:** The increase in circumference is $\Delta C = 25 - 20 = 5\text{ cm}$. * **Calculation:** Using the relationship $C = 2\pi r$, the increase in radius is $\Delta r = \frac{\Delta C}{2\pi}$. * **Calculation:** Substitute the value of $\Delta C$: $\Delta r = \frac{5}{2\pi}$. ### Exam Strategy & Shortcut Instead of finding the individual radii $r_1$ and $r_2$ and then subtracting them, directly use the difference. $\text{Difference in radius} = \frac{\text{Difference in circumference}}{2\pi}$. This saves significant calculation time. ### Common Pitfall A common mistake is forgetting the $2$ in the denominator and selecting $\frac{5}{\pi}$ as the answer, which corresponds to the change in diameter, not radius. ### Final Answer Therefore, the correct answer is **$\frac{5}{2\pi}$**.